Gorenstein projective dimension with respect to a semidualizing module
Copyright policy )Gorenstein projective dimension with respect to a semidualizing module
We introduce and investigate the notion of $\gc$-projective modules over (possibly non-noetherian) commutative rings, where $C$ is a semidualizing module. This extends Holm and Jørgensen's notion of $C$-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite $\gc$-projective dimension, showing in particular that they admit $\gc$-projective approximations, a generalization of the maximal Cohen-Macaulay approximations of Auslander and Buchweitz. Over a local (noetherian) ring, we provide necessary and sufficient conditions for a $G_C$-approximation to be minimal.
Final version, to appear in Journal of Commutative Algebra
13D02, 13D25, 13D05, Bass classes, $GCC$-dimensions, proper resolutions, Commutative Algebra (math.AC), 13D02, 13D05, 13D07, 13D25, 18G20, 18G25, Syzygies, resolutions, complexes and commutative rings, Homological dimension (category-theoretic aspects), semidualizing modules, bass classes, $G$-approximations, \(GCC\)-dimensions, FOS: Mathematics, Gorenstein dimensions, $C$-projectives, 18G25, Homological dimension and commutative rings, \(G\)-approximations, \(C\)-projectives, Homological functors on modules of commutative rings (Tor, Ext, etc.), totally \(C\)-reflexives, complete resolutions, Mathematics - Rings and Algebras, strict resolutions, Mathematics - Commutative Algebra, 18G20, complete \(PC\)-resolutions, Relative homological algebra, projective classes (category-theoretic aspects), totally $C$-reflexives, Rings and Algebras (math.RA), complete $PC$-resolutions, 13D07
13D02, 13D25, 13D05, Bass classes, $GCC$-dimensions, proper resolutions, Commutative Algebra (math.AC), 13D02, 13D05, 13D07, 13D25, 18G20, 18G25, Syzygies, resolutions, complexes and commutative rings, Homological dimension (category-theoretic aspects), semidualizing modules, bass classes, $G$-approximations, \(GCC\)-dimensions, FOS: Mathematics, Gorenstein dimensions, $C$-projectives, 18G25, Homological dimension and commutative rings, \(G\)-approximations, \(C\)-projectives, Homological functors on modules of commutative rings (Tor, Ext, etc.), totally \(C\)-reflexives, complete resolutions, Mathematics - Rings and Algebras, strict resolutions, Mathematics - Commutative Algebra, 18G20, complete \(PC\)-resolutions, Relative homological algebra, projective classes (category-theoretic aspects), totally $C$-reflexives, Rings and Algebras (math.RA), complete $PC$-resolutions, 13D07
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